No constant-mode hair in fixed-sector black-hole exteriors of flux-fixed general relativity

Branch-scoped theoretical note in the Gauged Constant Vacuum-Mode (GCV) framework. The note records a fixed-sector black-hole/no-hair compatibility consequence of the GCV source-assignment rule: in a connected, membrane-free black-hole exterior belonging to one fixed flux sector on the GR-exact GCV branch, the governed exact spacetime-constant vacuum mode cannot develop smooth independent radial hair. The proof is kinematic, using the topological equation dλ = 0 away from membranes. With no exterior local excitation sector activated, the exterior metric reduces to the usual Einstein solution with fixed Λ_eff, namely the Kottler/Schwarzschild-de Sitter family in the static spherical case. Apparent counterexamples are classified as ordinary local exterior stress, membrane-mediated sector jumps, or branch failure through no-leak/operator violations that promote the governed constant mode back into local dynamics. The record is not a new black-hole metric proposal, not a semiclassical evaporation analysis, not a singularity-resolution claim, not an information-loss analysis, and not a proof of the full upstream GCV framework. Its role is to supply the black-hole/no-hair compatibility entry for the GCV source-assignment and Governance Bridge route. Project homepage: https://johansson.digital